Fowler–Nordheim theory

7/28/2019 FowlerNordheim theory

1/10

MARCH 1 AND i5, 1942 PHYSICAL REVIEW VOLUME 51Electron Emission of Metals in Electric FieldsIIL The Transition from. Thermionic to Cols Emission*

EUGENE Gma aND CHARLES J. MULZ. IWtUniversity of Notre Dame, Notre Dame, Indnma

(Received October 14, 1941)The theory of Schottky emission is extended to include in the current those electrons which

tunnel through the top of the potential barrier at the metal surface when rather strong electricfields are used. It is found that these tunnelling electrons contribute to the periodic deviationsfrom the Schottky line, increasing the amplitude of the deviations. This increase leads to abetter agreement with experiment, especially for large fields. This agreement requires thatthe Nottingham reflection coefficient be very small for fields greater than 10' volts per cm.Expressions are developed for the electron current emitted by a metal for various field intensitiesranging from the small fields of thermionic work to the large fields used in cold emission.Results are obtained to indicate the temperature and field dependence of the electron currentfor all fields and temperatures of interest. Of particular interest is the expression developedfor the current in the "transition region, " i.e., T 500-1200'K and 7~10~-10 volts per cm.The modifications which must be made in the theory to take account of the polycrystallinenature of the emitting surface have no effect on the periodic deviations from the Schottkyline, and very little effect on the Ructuations observed in field photo-currents.

L INTRODUCTiON' 'T is possible to obtain electron currents emitted~ ~ from metals under widely varying experi-mental conditions. For example: the electronsmay be ejected from a hot cathode and collectedby fields with a rather wide range of intensities;or the electrons may be pulled from a coldcathode by the application of very intenseelectric 6elds. Despite the different experimentalconditions employed in these cases, a singlegeneral expression which is valid (under the usualassumptions of the free electron theory of metals)for all electron energies, 6elds, and temperaturesmay be given:

surface under the inhuence of the applied 6eld, F.Unfortunately, an evaluation of (1) valid for allenergies cannot be given in an analytical form;however, some special cases of interest andimportance can be fully treated. The familiarwork on thermionic and Schottky emission hasgiven a theory which holds for high temperaturesand low fields [high values of W in Eq. (1)],andthe well-known papers of Fowler and Nordheim'and of Nordheim' treat the case of very lowtemperatures and high fields Dower values of Win Eq. (1)]. In the present work an attempt ismade to extend these theories and to bridge thegap between the two theories. In line with this,

i = N(W)D(W, F)dW,Jo Ywhere lV is the energy which the electron has inthe velocity component normal to the emittingsurface, X(W) is the number of electrons in themetal with this energy which strike unit area persecond, and D(W, F) is the probability that oneof these electrons will escape through the metal~ This work is a section of a dissertation presented tothe Faculty of the Graduate School of the University ofNotre Dame, 1942, in partial fulfillment of the require-nents for the degree of Doctor of Philosophy.t Now at St. Louis University, St. Louis, Missouri.

Flu. 1. Energy diagram for electrons in and out of themetal. S' ' is the total height of the barrier at the metalsurface. Electrons from region A contribute to Schottkyemission, those from region 8 to cold emission, x2 and x3are the points at which @= TVV=O.' R. H. Fowler and L. W. Nordheim, Proc. Roy. Soc.119, 173 (1928).' L.W. Nordheim, Proc. Roy. Soc. 121, 626 (1928).


2/10

E. GUTH AN D C. M ULL I Nthe recent theory' of' the periodic deviations fromthe Schottky line is also extended to include theeRect of the electrons which penetrate the top ofthe potential barrier at the metal surface (Fig.1); this penetration effect is found to increase theamplitude of the deviations so as to improvesomewhat the agreement between theory andexperiment at the higher field intensities used. Itis found that the improved agreement betweenthe theoretical and experimental deviations re-quires that the reHection coefhcient proposed byNottingham4 decrease with the applied field insuch a way as to become nearly zero for fields ofthe intensities used to obtain the periodic devia-tions. An interesting result is also found for thecurrent emitted when rather high fields andintermediate temperatures are used. In Part IIof the present paper, the results which can beobtained from Eq. (1) for the case of an idealsurface with a single "effective" work function,or for a single crystal face, are discussed; inPart III the effects of the polycrystalline natureof the emitting surface on these results are con-sidered; and in Part IV a brief derivation forthe currents obtained under the various condi-tions discussed in II is given.II. DISCUSSION OF THE RESULTS FOR AN IDEAL"SINGLE WORK FUNCTION SURFACE"OR FOR A SINGLE CRYSTALIn order to evaluate (1) it is convenient tosplit the integral up into parts corresponding tovarious ranges of the energy 8". If only values ofW greater than W, ' (Fig. 1) are considered, that

is, if the field applied is not strong enough tocause an appreciable number of electrons withenergies below W, ' to be emitted, Eq. (1) leads toan expression for thermionic or Schottky emis-sion; if only values of W less than 8'; are con-sidered, that is, if the filament temperature is solow that only a negligible number of electronshave energies greater than W;, Eq. (1) leads to anexpression for cold emission. For values of R'ranging from TV; up to 8' ', the current graduallytakes on a stronger temperature dependence andweaker field dependence, passing from cold emis-sion to thermionic emission; however, an ana-lytical expression for the current valid in the' E. Guth and C. J.Mullin, Phys. Rev. 59, 575 (1941).4W. B.Nottingham, Phys. Rev. 49, 78 (1936).

entire region W,V8'' has not yet beengiven. In the following work some limiting casesare treated.

W&f 1 cos v~i 1 W.'}2kT 4 4xo' & ([sn+(1/kT)]'+p'n'}&cos vs+ (2)

I [2irn(1/k T)]'+p'n'}&where atomic units of length (a=k'/me' =0.527A)and energy (unit=e'/2@=13. 67 ev) are used, andn=(2xo')&; p=2xnkT; xo=cha, r. Schottky dis.tance (Fig. 1):xo= [(300e)&/(2 X0;527) j(108/7&)F=volts/cm; y =work function for zero fieldfor field F: x' =y(1/xo); W ' =W;+ y'P = y+2 ln 2; y=Euler's const. =0.5772B= 160m'm'e%k'

442 2 +tan ' an ' en+ (1/k T)4N 2

vf, xo& +tan ' tan 'w. ~ m.n(1/kT)If the relatively small periodic term is neglected

(2n+1)pi=B(kT)'e &'"r Q (1)"- (3)n (n+ 1)ii'+ iiIf only the first term of the summation is takenthis result reduces to one obtained by Bethe~

If p&)1, which means that high temperatures and~A. Sommerfeld and H. Bethe, IIaedbuck der Pkysijf;(1933),Vol. 24/2, p. 442.

(a) Low Fields, High TemperaturesIn this case only electrons with energies abovea certain value 8 contribute to the current,the others being reHected back into the metal

when they strike the surface; the contributingelectrons are those from region A in Fig. 1. Thecurrent from this region is(2n+1)pi=B(kT)'e "'" P (1)=o n(n+ 1)p'+ ii 1


3/10

ELECTRON EM I SSION OF METALS 34ilow fields are employed, Eqs. (3) and (4) reduceto the familiar equation for Schottky emission, inwhich the mean transmission coefficient 5 1,i =8(AT)'e-&'Ibr.It is seen that (5) and (3) differ in that (5) effec-tively considers only those electrons which haveenergies greater than 8' ', i.e., which surmountthe potential barrier of Fig. 1,while (3) also takesaccount of those electrons which tunnel throughthe barrier in the region 8",'b~ 8"~W '. Thistunnelling occurs even for rather weak fieldsbecause even then the transmission coefficient isnot zero at the top of the barrier of Fig. 1. Acomparison of Eq. (2) with the equation obtainedfor the current by neglecting the tunnel e8ectshows that the second periodic term (cos nb) of(2) arises from the electrons penetrating thepotential barrier at the metal surface. An inter-esting fact about the periodic term in (2) is thatit becomes small both for low and high fields; forsufficiently high fields xbxi 1/2Wso that thecoefficient (1/4xb2W, ') becomes smaller as thefield becomes very high. The contribution of thesecond periodic term due to the tunnel e6ectgives a better agreement between the theoretical

and experimental~' amplitudes of the deviationsat the high end of the field range used.With a.ll electrons accounted for in Eq. (2), theperiodic deviations have an amplitude in goodagreement with observation. This agreementaRords a good opportunity to observe whether ornot the electrons reaching the metal surface aresuffering, at these rather high field intensities, areAection of the type suggested by Nottingham. 'In order to explain the non-Maxwellian distri-bution of the slow electrons in thermionic emis-sion, Nottingham proposed that the electronssuffered a reflection

Rge&~~ ')'", co 0.2 ev. (6)However, as Dr. Nottingham pointed out to theauthors, the velocity distribution experimentsrequire only that co 0.2 ev for the very low fieldsused in those experiments. A transmission coeffi-cient which meets the conditions imposed by R~and at the same time introduces some of theinfluence of the barrier of Fig. 1 is given by

D =D.(1Ry),where D, is the transmission coeScient for thebarrier of I'ig. 1. Kith this transmission, thecurrent from the region A becomes

1 1 !g(lcT)2s g'IkT J Q ( 1)n!~=o Ln:abc+1 nic+1+(kT/cd) )

whereW&~ 1 cos Qy cos Q2!1 (8)2kT k 4xb'W ') ILxa+ (1/k T)j'+P'n'}l IL2xa+ (1/k T)+ (1/(y) j'+P'n'}&

4&2 2 8' 'I,= xo~ +tan-'S N.~ an+a+ (1/k T)+tanw.~ an '~a+ (1/k T)+ (1/cb)

To keep the amplitude of the deviations appreci-able wouId require thatn.n 1/co, or ce 1/2000 atomic units, cd 1/200 ev.Thus, co must depend upon the field in such a waythat ao 0.2 ev when I' 0, and au~0 as F takeson the rather large values needed to obtain theperiodic deviations. Practically, Eq. (8) shows

that the Nottingham reRection does not operateat these relatively high fields. A discussion of theNottingham reRection eersgs the theory of emis-sion from "patches" is given by Nichols. '6 R. L. E. Seifert and T. E. Phipps, Phys. Rev. 56, 652(1939).' D. Turnbull and T. E. Phipps, Phys. Rev. 56, 663{2939).%.B.Nottingham, Phys. Rev. 57, 935 (2940).' M. H. Nichols, to be submitted for publication.


4/10

E. GUTH AN 0 C. MULLI N

08

region 8, since the transmission coef6cient forthis region decreases exponentially with de-creasing energy. The current from this region isthen,

0.6X=8 exp 8x02x~ 2682x04X

0,504

( 1)++1+ (kT)' (9)(n+4xp'Ox&k T05o.p.

I I l l5 6 7 8 9 lO I I l2 I5Fxl0

or, if we express x in electron volts and Ii in voltsper cm,P2 ( 1)nesti= 1.55X10-' +120.5T' P2X x=1 g

Frr. . 2. The Nordheim function 8 plotted against theaccelerating field I.It may be noted that the value ~=0.2 ev leadsto a mean transmission coefficient D=-'whilethe value ra 1/200 ev leads to a D 095.In the case of photoelectric emission under the

inAuence of 6elds high enough to cause pene-tration of the topmost part of the barrier, thepenetrating current plays a much less importantrole than in the above cited case of Schottkyemission; the reason for this is immediately clear:In the photoelectric case the transmission de-creases exponentially as the electron energydecreases from 8',', but the number of electronsavailable remains almost constant (since theseelectrons have energies less than W~); hence, thecurrent, which again depends upon the productD(W, F)X(W), decreases very rapidly as Wbecomes progressively less than t/I/' ', this quickdecrease makes the contribution to both non-periodic and periodic parts of the currentunimportant.

Xn+8 81X 10+'Ox*T/Fexp (.838X10'Ox&/F) amp. /cm' (10)

where 8 is a slowly varying function' of I' whichvaries from 2 to 0 as I' varies from 0 to ~. e isplotted as a function of F in Fig. 2. If thesummation term is neglected completely (Thiscorresponds to taking only the 6rst term in theexpansion of the Fermi distribution function, avalid approximation for low temperatures), Eq.(10) reduces to

P2i =2.55X20 ' 'xexp (6.838X10'Ox&/F) amp cm'. (11)

This formula has been recently checked experi-mentally by Haefer. " By taking the sum intoconsideration one obtains the temperature de-pendence of the current from region 8 of Fig (1);.(h) High Fields, Low Temperatures

At low temperatures there are very few elec-trons with energy greater than TV;. However,with su@ciently high values of the applied field,electrons with energies less than W; are pulledfrom the metal. The electrons contributing tothis cold emission are from the region J3 of Fig. 2,most of them coming from the uppermost part of

'0This function was introduced by L. W. Nordheim,reference 1, to determine the eHect of the image force oncold emission.' R. Haefer, Zeits. f. Physik 116, 604 (1940). Thisauthor determined the actual field strength acting on thepoint of the field emitting metal by obtaining the curvatureof the point from electron microscope investigations. Hefound that field emission sets in at about 3X10~volts percm as required by Eq. (11).Thus no "sensitive spots"need be introduced in order to explain the differencebetween the empirical and theoretical thresholds. Haeferaccurately verified the dependence of the current on x& asgiven by (11).


5/10

ELECTRON EM ISSIQN OF METALSif T is small so that 8.81X1Py1VT/F1, Eq. (10)gives for the temperature dependence

i(T) XT' - (1)"+'=1+7.77X10'O' Q . (12)i(0) P& st-iThe first term of the series in Eq. (12) has beengiven by Bethe" who points out that this resultis in agreement with the experimental data ofMillikan and Eyring. "The temperature depend-ence is of the same type as that obtained f'orphoto-emission for the simple reason that theelectrons in each of these cases originate in thesame region B.For low temperatures the T' term in (10) isunimportant, but it may become appreciable forhigher temperatures; in any case, ii seems thatthe inclusion of the temperature term in (10) isinconsistent unless this term is modi6ed toinclude the contribution of those electrons which,because of the temperature dependence of theFermi distribution function, have energies greaterthan t/V;. However, field emission for which thetemperature term is large (say for T=800-1000'K, F=24X10' volts/cm) is not strictlycold emission, but rather may be classed as atransition case between cold and thermionicemission. The current obtained under these"transition conditions" is discussed in Section (c).(c) Intermediate FieMs and TemperaturesIn this case the emission from region C of' Fig. 1must be considered. The current from this region

should depend upon both the temperature andthe 6eM. An expression valid f'or all of region Cis not easily obtained, but the contributions fromthe upper and lower parts may be given. A par-ticularly important case involving the currentfrom this region has been mentioned in the previ-ous section. If the temperature and 6eld are suchthat 4xo'~&OkT &1 (atomic units),i.e.,8.813X10'Oy&T/F&1 (ev and volt/cm) (13)

the current from region C is

i=Bk'T' expSxp'

ex~3( 1)n+1 (14)nxp'8y&k T

By adding this current to that from the region 8,the total current for the fields and temperaturessubject to the condition (12) is obtained(1)"+'+k'T' Q3 9 xp'g S

Sxp' 1i =8 exp ey& 161 1

x! + (15)kn+4xo'Oy&kT nxo'Oy&kTJor, putting y in ev and F in volt/cm

F2 ( 1)++1i= 1.55X10 ' +120.5T' Q6'g

6T&f000 K

2'oO 02

4

x!'In+ 8.813X10'OX~T/F1

!".813X10'OX~T/F~Xexp L.838X10'6"1/F] amp/cm'. (16)Experimentally it should be possible to make athorough study of currents under these "transi-tion conditions. " If T= 1000'K, say, then bychoosing y=4.531 ev (the value for tungsten)(16) is valid if F~& 1.8X10' volts/cm. A plot of

A. Sommerfeld and H. Bethe, Ejundbggk der Ekysik{j933),Vol. 24/2, p. 442.'3 R. A. Millikan ctnd C. F. Eyring, Phys. Rev. 2f, Sj.{.926).4 ~5Fs lo '

FIG. 3.A plot of the current given by Eq. (16).


6/10

E. GU TH AN 0 C. M U LL I Nthe current obtained from (16) is given in Fig. 3.It may be noted that for fields near the thresholdfor emission, Eq. (16) gives a current many timesthat expected from the simpler Nordheim ex-pression given in Eq. (11). In particular, ifT=1000'K and F=2X10' volt/cm, (16) givesa current about five times as large as that ob-tained from (11). If F=3X10', (16) yields avalue about 1..7 times as large as that given by(11).The larger current is due chiefly to thoseelectrons coming from region C.The expression (14) for the current from thelower part of C shows, as was expected, a largertemperature dependence and a smaller 6elddependence than was had for cold emission.Photoelectric emission could be obtained fromthe region C by applying 6elds high enough tomake the electrons struck with low frequencyphotons tunnel the barrier in the region adjacentto C. The photo-current from this region isi=exp L6.838X 10'8(xv)&/F]

F2 ~ ( 1)n+1~.55X~0-' +pep(gv) ~=& Ik2T2X- (17)n+8 81X10.'e(yv)&T/F

This current is very similar to that for cold emis-sion as given by Eq. (9). However, the photonshave the eRect of reducing the work function byan amount hv, so that really the electrons fromregion 8 are effectively raised to region C andtunnel through the barrier from this region.

HI. THE EFFECT OF PATCHES ONTHE EMISSIO'In a11 the above work the electrons are assumedto be emitted from a surface with one uniform"effective" work function. If, instead, the current

is assumed to come from a number of crystal faces(hereafter called patches) with different workfunctions, the previous expressions for the cur-rent must be modified. If the applied 6elds usedare high (xppatch diameter 100A) as is"For a complete discussion of the theory of patchemission see M.H. Nichols, reference 9.cf.also C.Herring,Phys. Rev. 59, 889 {1941).%e are indebted to Drs.Herring and Nichols for letting us have the manuscriptsof their papers before pubhcation.

WoN0

Q W&-oxx~O

FIG. 4. The potential barrier at the metal surface if theimage force is neglected.generally true above, each patch may be assumedto emit independently, so that the resultingcurrent is i=elfi., (18)where iis the current from the nth crystal face,and f is the fractional area of this patch.

i =P fi=Pf&oDT'e '"&'"r-(19)n=l n Iand from the second

f i e blkT Q f g DT2e (g ky)Ikr (20)If 8&kT, i~&&ii so that the principal contributionto the current comes from the patches of lowestwork function. In this case the "ideal surface"emission constant +Ap must be replaced by

Apg fn=1which is always less or equal to A0, the equalityholding only if the entire surface has the samework function. For 6elds high enough that thepatches emit independently, the Schottky line isthe same as for an ideal surface, i.e.,

ln i(F)n i(0) Ax/k T= e&F&/k T. (21)If the various groups of crystal faces have workfunctions which differ by an amount 5)k'1, thecurrent comes almost exclusively from the facetsof lowest x, and the periodic deviations found inPart II are unaffected. However, if there exists a5 for which the condition b&k 1is not fulfilled,

(a) Thermionic EmissionIf it is assumed that a certain number p ofpatches have work functions very nearly equal,the work function of the group being called x, andif g patches have a work function x+6, thecurrent from the first group is


7/10

ELECTRON EMISSION OF METALSthen a current contribution from the corre-sponding group of patches must be considered.However, a 8 this small would not affect theperiodic deviations of Part I I (which dependupon W,), since their dependence upon x is arather insensitive one, and the only eft'ect of thispatch group is to increase the current. Thus, itis seen that the periodic deviations are neverdestroyed by interference" between the emis-sions of difkrent patches.The arguments applied above about the e8ectof patches on thermionic emission may be applieda,lso to cold emission (here there is no periodicterm, of course) which, according to Eq. (9)depends exponentially upon x&.

1 ~ (1)"+'i. Bk'T' +'P e "'6 2 -1 n' (22)kvy.+Ax&0,

or( 1)a+1Bk'T' P e"' if 8&0.n 1 +2

(1) Photoelectric EmissionIn the case of photoelectric emission withrather high 6elds, neglecting the small periodicterm we have

It is seen that in this case the current does notfall o8 exponentially with increasing work func-tion. Again suppose the two groups of patcheswith work functions x and x+6 as above. Here itis possible to let b have an appreciable value andyet to obtain an appreciable current contributionfrom the patches of work function g+b. Sincethe phase and amplitude of the periodic devia-tions found in the photo-current" emitted by ametal are dependent somewhat upon the workfunction of the metal, it is possible that the inter-ference of the several currents from the crystalfaces of diferent work functions might givedeviations somewhat diRerent in character fromthose calculated for an "ideal, " single workfunction emitting surface. However, this inter-ference e6ect must be small, for in order thattwo currents from patches with diferent workfunctions may have a large interference eR'ectin their periodic terms, the two patches musthave a rather large di6'erence in their workfunctions; but a large difference in work func-tion means that one current is small in corn-parison with the other, so that the interferenceeffect will be small. This holds in spite of thefact, already pointed out by Nichols, that theFowler plots are modi6ed by patches. In the caseof the Fowler plot a change of the work functionby 8 is of importance, while the periodic devia-tions, depending on W, ( 10 ev) are ratherinsensitive to changes in x of a few percent.

1V. DEMVATION OF THE RESULTS(a) Low Fields, High Temperatures

In this case Eq. (1) is evaluated, (a) for W, '~W~, and (b) for W, 'WW, ', these twoontributions are then added to give the total current from region A.(a) W,'W~.

For this case the transmission coefhcient and energy distribution are given by'61 W,&~ 1 i exp[ma(WW, ')jcosu,W)= 1+exp [mn(WW ')g 2 k 4xo~Wo2& (1+exp [ma(WW ')]}&

xo& +tan-~ n(WW.')3 8",&X(W) -BkT exp [(WW~)/kT), F.Quth and C. J.Mullin, Phys. Rev. 59, 867 C&94&).'6 For these results and the calculation of the corresponding current cf. E. Guth and C. J. Mullin, Phys. Rev. SQ,STS (I941). The transmission coe%cient given here is obtained by use of the W'.K.B. solution to the wave equation.


8/10

E. GUTH AN 0 C. M ULL I N00 y 00i = D(F, W)N(W)dW= D(F, o)N(o)do,

o=WW'.=WW.+1/xo,1 Wo& [11/4xp'W, ')] cos s.i =Bk'T' exp ('/kT) P (1)"~=o 2norakT+1 2kT {[ora+(1/kT)]'+Poa'}~ (25)

v.= xo& +tan ' an '4 ora+ (1/k T)(b) W 'WW,'.

The 6elds in this region are presumed to be such that the electrons penetrate only the topmost partof the barrier of Fig. 1.Then proceeding as in (a)exp [ora(W, 'W)]D(F, W)= 1+exp [ora(W. 'W)] 2 {1+exp[ora(W, 'W) }&Wo~ [11/4xo W )]exp [2ora(Wo' W)cos uo, (26)

4&2Ng Xp&3

2 W&+tan-' +Pa(W. 'W),8'& 4N(W) BkTe '~ ~'&I'r, - (27)

~ ling, ' ~00N(W)D(W, F)dW= ~ N(y)D(y, F)dy,~S'o~ a 0y =W,'-W= W,W-1/xp,

oo 1 W.& (11/4xo'W, ')i =Bk'T'e &'"r g (1)"~-o 2(n+1)orakT 2kT {[2ora1/kT)]'+P'a'}~4&2 2 W~ Pa

pg = xp~ +tan ' +tan3 W, & 4 2ora(1/k T)cos no, (28)

The region of integration is taken as O~y~ since it is assumed that the integral vanished at theupper limit. Actually, the integration range should be 0 corresponding to W, 'WW '.Adding the results of (a) and (b) gives for the current from the region A

(2n+1)pi =Bk'T'e x'"r Q (1)"n(n+ 1)u'+ uCOB Sy.~ ( COS V~W.o } +

2kT k 4xo' I {[ora+ (1/k T)]'+P'a'} {[2ora1/k T)]'+P'a'}(29)

If the transmission coeScient given by Eq. (7) is used and only electrons for which W&~ W, arecountedi=BkTe x'Ior~t D,(1 'I")e 'I" do

0

=BkTe &'"r)t D,e 'foldo0

fs 00jg tp,(1/k T)+(1/su)pod p


9/10

ELECTRON EM ISSION OF METALSThe 6rst integral has been evaluated aboveand the second integral may be obtained from thefirst by replacing 1/kT with 1/kT+ 1/oo. Hence

1 1 !=8k'T'e x'"r Q (1)"{.ni+1 np+1+ (kT/oo) )W( 1 cos Qy COS N2W. ! (30)2yT 5 4x o ) {[KA+(1/)t(T)]2+p2A2} {[KA+(1/pT)+(1/N)]2+po&2} l

u~ xo&w.~w.~an ' an 'ma+(1/kT)

4v2 2 W. Pngp& an ' an3 W, 4 sn+ (1/k T)+ (1/&o)(b) High FieMs, Low Temperatures

For this case only electrons from region 8 need be considered. Here1 xyD exp (2Q); Q= { W,W ! dx.2x 2x,o) (31)

The effect of the image force on electrons from region 8 is small. Hence, to evaluate Q it is legitimateto set 2Q=28Qo, whereX y& 8X22Qo { W W ! dx= (WoW)2g 2j

is the exponent which is obtained in the transmission coeScient if the eAect of the image force isneglected, and 8 is the correction function which has been used previously. The potential barrierused in the determination of Qo is shown in Fig. 4. Since the transmission coefficient decreases expo-nentially, the current vanishes when =W;W is large. Hence, Qo may be expanded in powers of :and since Qo- (4xo'/3) x+2xo'x " (-1)"+'X()=B +AT Q e "'"r

8XO2 (-1)+' t" - ( n q-exp [ 48xo'x]d+kT g ~ exp { 4o'x+~o n JL I TJ8x02x~ 1 w (

1)"+'+Z1682@0&g ~=& n $2T2n+4X02ey &k 1 (32)P2 ( 1)++I+120.5T' Q9'x n 1,~= 1.55y10-~

or if x is put in ev and F in volt/cmn+ 8.81X10o(8xT/F)

Xexp [6.838X10~8x/F] amp/cm'. (33)


10/10

E. GUTH AND C. MULL IN(c) Intermediate Fields and Temyeratures

If the fields and temperatures are such that 4xp'Ox&kT&1, the current comes from the lower partof C. Again if we take D-exp I -(8x"!3)Bx~+4I'x"x9j, f3=W W-;as above, and use ( l )n+1N(P) =BkT Q e "P'"rn=1there resu1ts

t =Bk2T2 exp 8x 2 " ~ ( ])p+1 ~wBX& Q exp 4xp2BX'*(n/kT) jPdP3 n=1 yg aJ p=Bk2T2 exp 8xp' (1)"+'Bx~ 23 n=i n nexp2y. &kT

By adding this current to that from region 8, the total current for the condition 4xp20y&kT&1 isobtained

i=8 exp 1682xp4y (1)"+' f' 1 1

!k2T2 g=1 n En+4Bxp x&kT n 4Bxx&kTJ (35)or putting x in ev and F in volt/cm

Pp ~ ( l)p+1 ( 11.55X 10 ' +120.54T' Q !+Bpx -1 n (n+8.81X10'(BX&T/F) n8.81X102(BZ&T/F))Xexp [6.838X10'(Bx~/F) ] amp/cm'. (36)

The current penetrating the very top of the barrier has been given in Part IV (28).The photoelectric emission from C consists of electrons for whichD(W+hv, F)~ exp 80xp' (XV) & 4BXp2(XV)&! 13

(1)"+'X(W)=B p+kT Q e "'"r .Because of the large exponential in the transmission coefficient, the majority of the current comesfrom electrons for which y= t/t/'; V is small; hence

z=B exp Sxp'e (xv)& I 1682xp4x (1)++1+O'T' Qn=1 n+4Bxp'(x hv) &kT(3&)

or, using ev and volts per cm,Ii2 ( 1)++1~= i. .55yj.0-6 +120.5T' QB2(Xv) =1 n n+8.81X 102[B(xv) &T/Fj

S(xv) &.exp .838X 10' amp/cm'. (38)

Fowler–Nordheim theory

Documents

Transcript of Fowler–Nordheim theory